10 – pp-chain and CNO cycle

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 1 pp chain reaction in the Sun

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 2 10.1 - Hydrogen burning – ppI chain As a star forms from a giant cloud in the interstellar medium (protostar), density and temperature increase in its center. Fusion of hydrogen (1H) is the first long term nuclear energy source that can ignite. With only hydrogen available (for example, a first generation star) the ppI chain (see previous slide) is the only possible sequence of reactions. (other reaction chains require catalyst nuclei). The ppI chain proceeds as follows:

2 Step 1: p + p → He This does not work because 2He is unstable + (10.1) p + p → d + e + ve

3 (10.2) Step 2: d + p → He d d 4 He This does not work because the d abundance is + → too low. d + p leads to rapid destruction of d.

3 4 Step 3: He + p → Li Does not work because 4Li is unstable 3 He + d → 4 He + n Does not work because d abundance is too low 3 He + 3 He → 4 He + 2p (10.3)

Last reaction is OK because Y3He gets large as there is no other rapid destruction. introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 3 10.1.1 - S-factors for the pp chain All reactions in the pp-chain are charge-dependent. The cross section has the form given by Eq. (9.49) which can be cast in the form b where 1 − e E A A σ ∝ (10.4) 1/2 A = 1 2 E b = 31.28 Z1Z2A keV (10.5) A1 + A2 (10.6) Typical units for S(E) are keV barn.

As we have written in Eq. (9.55), the reaction rate samples very low energy cross

sections, or very low S-factors, through the Gamow Window E0 energy being very small. Then the integrand below around the Gamow window can be approximated by a Gaussian with energy width ΔE:

E # b E & − -% + ( 8 −3/2 kT 8 −3/2 $ E kT ' < σ v >= (kT) ∫ σ (E)Ee dE = (kT) ∫ S(E)e dE πµ πµ where (10.7) 2 " b E % 1" E−E % 4 1/6 −$ + ' − $ 0 ' 2 2 5/6 kT 2$ ΔE/2 ' ΔE = E0kT = 0.23682 (Z1 Z2 A) T9 MeV e # E & ~ e # & 3

(10.8) (10.9) which leads to (9.55) after integration, as proven in the next two slides. introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 4 S-factors at very low energy The standard method for estimating such an integral is to find the energy that maximizes the exponential and expand around this peak in the integrand. This corresponds to solving ! $ 3/2 d # b E & 2E # + & = 0 ⇒ b = 0 (10.10) # & dE " E kT % kT

Now carrying out a Taylor expansion around E0:

! $ 2 # b E & ! df $ 1 2 ! d f $ f (E) # & ~ f E E E E E = + 0 + − 0 # & + − 0 # 2 & +! (10.11) # & ( ) ( ) ( ) " E kT % "dE %E 2 "dE % 0 E0 ! $ df 1 2 # & = 0 ⇒ f (E) ~ f E + E − E f ''(E ) (10.12) "dE % ( 0 ) ( 0 ) 0 E0 2

1 2 - E-E f ''(E ) Thus 8 -f E ( 0) 0 < σ v >= (kT)−3/2 S(E)e ( 0)e 2 dE πµ ∫

1 2 - E-E f ''(E ) 8 -f E ∞ ( 0) 0 ~ (kT)−3/2S(E )e ( 0) e 2 dE (10.13) πµ 0 ∫ 0 introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 5 S-factors at very low energy

-f E ( 0) 4 −3/2 e < σ v >= (kT) S(E0 ) (10.14) µ f '' E ( 0 )

3b 3 But E0 b 3E0 f ''(E ) f (E ) and 0 = 5/2 = 0 = + = 4E 2E kT kT E kT 0 0 0 (10.15) (10.16) Thus, we get Eq. (9.55) as promised:

! $2 16 1 1 3E0 < σ v >= S(E0 )# & 9 3 µ 2παZ1Z2 " kT %

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 6 S-factors at very low energy

Better (and this is often done for experimental data) one expands S(E) around E = 0 as powers of E to second order: 1 S(E) = S(0) + ES'(0) + E2S''(0) (10.17) 2

If one integrates this over the Gamov window, one finds that one can use

equation of previous slide when replacing S(E0) with the effective S-factor Seff

' 5 S'(0) ! 35 $ 1 S''(0) ! 2 89 $* Seff = S(0))1+ + #E0 + kT&+ #E0 + E0kT&, ( 12τ S(0) " 36 % 2 S(0) " 36 %+

3E0 (10.18) with τ = and E0 as location of the Gamov window. kT

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 7 The ppI chain

Let us start by looking at the nuclear chart Z 3He 4He 2

In our notation, we call this as a 1H 2H “chain” of reactions, or 1 reaction chain: 0 1

N “bottle neck”

(p,e+) (p,γ) (3He,2p) 1H d 3He 4He

The first reaction in the chain is p + p à deuteron (d) + e+ + ν. The cross section for it is very small. The other reactions have to wait until this reaction occurs. That is why this reaction is called a “bottleneck”.

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 8 The p + p à d reaction The p + p à deuteron (d) + e+ + ν cross section is very small because the protons have to overcome their mutual repulsion, tunnel to a distance of about 1 fm. Then one of them has to decay to neutron + neutrino and the neutron has to be captured by the proton to form the deuteron. The cross-section for this reaction for protons of energy below 1 MeV is very small, of the order of 10−23 b. In fact, the effective energy for this reaction in the Sun is about 20 keV and this value is much lower.

Another possible reaction forming the deuteron is p + e- + p à d + n, but the frequency with which it occurs is about 400 times smaller than the pp reaction.

-25 The S-factor for the pp reaction is Spp(0) = 4 x 10 MeV b and the reaction rate for it is found to be

9 # & 1 2 1.15×10 2 2 3.38 3 1 r = n σ v = X ρ exp%− ( cm− s− (10.19) pp 2 p pp T2/3 % T1/3 ( 9 $ 9 ' where X is the hydrogen mass fraction. The higher order correction leads to -1 S’pp(0) = (dSpp/dE)E=0 = 11.2 × Spp(0) MeV . See Reviews of Modern Physics 85, 195 (2011).

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 9 The p + p à d reaction The rate of change for proton number because of the pp reaction is given by the

dn p n p = − (10.20) dt τ p where τp is the proton half-life due to the reaction. The lifetime is then

n p n p (10.21) τ p = − =

dn p / dt 2rpp where the factor two is because two protons are destroyed in the process.

-3 Using T9 = 0.15, X = 0.5 and ρ = 100 g cm in Eq. (10.19) and using Eq. (10.21) one 9 gets τp ~ 5 x 10 years. This is a reasonable estimate for the lifetime of the sun.

Note: it is customary to call Spp by S11, meaning A = 1 for each of the two protons. For example, the S-factor for the pd reaction is called S12.

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 10 p + d à 3He + γ

The S-factor for this reaction is

S12(0) = 0.241 eV b.

In the figure we show the world data (as of 2011) for the S- factor for this reaction. The curves are theoretical calculations and/or experimental fits. The band is associated to the experimental error bars.

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 11 3He + 3He à 4He + 2p The S-factor for this reaction needs to include higher derivatives of S for a better precision of the reaction rate calculation, as in Eq. (10.17).

In the figure we see that the laboratory screening causes the S- factor to deviate from the desired “bare” S- factor. It is thus necessary to include screening corrections. The table below summarizes the fits for this reaction

Rev. Mod. Phys. 85, 195 (2011).

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 12 10.2 - Reaction networks with a bottle-neck

Consider a schematic chain of proton captures shown below:

(p, γ) (p, γ) (p, γ) (p, γ) 1 2 3 4 Slow bottle neck

We will assume that:

• Y1 ~ const because the depletion is very slow due to the “bottle neck”

• Capture rates are constant (Yp ~ const because of the large “reservoir”, abundant protons in the environment) The abundance of nucleus 2 evolves according to the equations

dY 1 2 (10.22) λ = Y ρ N < σ v > (10.23) = Y1λ12 −Y2λ23 12 1+δ p A 1→2 dt p1

The first term on the rhs of Eq. (10.22) is due to production of 2, the second term due to destruction of 2.

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 13 Reaction networks with a bottle-neck

We have assumed that Y1 ~ const and Yp ~ const. Hence, after some time Y2 will then reach an equilibrium value regardless of its initial abundance. When equilibrium is reached, we get

dY2 = Y1λ12 −Y2λ23 = 0 (10.24) and Y λ = Y λ (10.25) dt 2 23 1 12 When an equilibrium condition of this form is reached we call it steady flow.

At a later time we get a similar equation for Y3 : dY 3 = Y λ −Y λ = 0 (10.28) dt 2 23 3 34

and Y3λ34 = Y2λ23 (10.29) leading to result for Y 2 Y3λ34 = Y1λ12 (10.30) and so on …

In steady flow Yiλi i+1 = const = Y1λ12 (10.26) or Yi ∝τ i (10.27)

That is, all abundance Yi are proportional to the i species destruction rate λi,i+1.

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 14 Approaching equilibrium −t/τ 2 Y2 (t) = Y2f − (Y2f − Y2i )e For λι ~ const the solution of Eq. (10.22) is (10.28) where Y 2 i is the initial abundance and Y 2 f the final (equilibrium) abundance. Regardless of the initial abundance, the equilibrium is reached exponentially within a time equal to the lifetime of the nucleus.

Application to the ppI chain (10.29) Ydλd+p = const = Ypλp+p “bottle neck”

(p,e+) (p,γ) (3He,2p) 1 3 4 large reservoir H d He He (Yp ~ const)

Hence, the equilibrium Yd is extremely small 1 -18 Y N v (~ 4 x 10 in the sun). In the sun, the Y λ p ρ A < σ >p+p d p+p 2 equilibrium is reached within a time set by = = -2 3 Y Y N v NA <σv>pd = 10 cm /s/mole, which yields p λd+p pρ A < σ >d+p a deuteron lifetime of

< σ v > -19 Yd p+p S = 3.8 x 10 eV b 1/ (Y N v ) 2s = τ d = pρ A < σ >p+d = Y 2 < σ v > S = 0.24 eV b p d+p (10.30) (10.31) introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 15 3He equilibrium abundance (p,e+) (p,γ) (3He,2p) 1H d 3He 4He

In this case two identical particles fuse and therefore the destruction rate λ3He+3He is not constant. 1 (10.31) λ3He+3He = Y3HeρNA < σ v >3He+3He 2

This depends strongly on YHe itself. The equations cannot be solved with simple approximations and have to be solved numerically. The result is that the 3He abundance is much higher than that of d. Therefore, the fusion 3He + 3He is very probable and 4He can be formed in this way. The ppII and and ppIII chains If there is enough 4He in the medium, it will be a catalyst of the ppII and ppIII chains to produce even more 4He. 4 - 4 ( He,γ) (e ,ν) (p, He) ppII (14%) 3He 7Be 7Li 4He

ppIII (0.02%) (p,γ) (β+) decay 8B 8Be 2 4He

These chains also have to be solved numerically. introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 16 Electron capture on 7Be followed by beta decay

A curious and different process occurs in the ppII chain when an electron is captured by 7Be. After the capture of an atomic electron, a neutrino is emitted . The total Q-value for the process is

QEC = - 862 keV followed by Qβ+ = QEC + 1022 = 160 keV, thus becoming energetically possible.

In summary, the timescale of the pp-chain is dominated by the ppI chain. However it produces only ½ of 4He per pp chain. The ppII I and ppIII produce the other half thus doubling the production of 4He. II III

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 17 10.3 - The CNO cycle The CN cycle is part of the CNO cycle. It is the left branch I of the CNO-cycle shown in the figure. The CN cycle was proposed by Hans Bethe in 1938 as a way to use 4 protons to produce 4He, thus gaining substantial energy per cycle (~ 26 MeV).

The CN cycle is a reaction chain using 12C and 14N as catalysts. Their nuclei remain at the end of the cycle. The 12C nuclei act as seeds that can be reused, despite their very small abundance compared to that of hydrogen.

The CN cycle involves the reaction chain

12 13 + 13 14 15 + 15 12 C(p,γ) N(e νe) C(p, γ) N(p,γ) O(e ν) N(p,α) C

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 18 The CNO cycle The CN cycle can leak from the reaction 15N(p,γ)16O and form another branch, known as the ON, or CNOII cycle. The nitrogen returns to the initial CN cycle as a 14N nucleus through 17O(p,α)14N. This is shown in the figure of the previous slide. The ON cycle only works 0.1% of the time compared to the CN cycle because the S-factor for 15N(p,α)12C is 1000 times larger than that for 15N(p,γ)16O. A much rarer process happens at higher temperatures, adding a CNOIII cycle through the 17O(p,γ)18F reaction. All branches of the CNO cycle are shown below.

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 19 p-p chain vs. CNO Cycle The CN cycle operates at higher temperature than the sun. It is relevant for temperatures T6 > 20. But the branches II, III and IV can occur at much larger temperatures (T ∼ 108 − 109 K such as (a) hydrogen burning at the accreting surface of a NS, (b) explosive burning on the surface of a WD (novae), (c) in the outer layers, during the shock wave of a supernova explosion. For the sun the pp- chain burning predominates as shown by numerical calculations shown in the figure .

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 20 Solar fusion: Summary A detailed account of solar fusion reactions with cross sections and S-factors for all relevant cases has been reviewed in the article below by an international task force.

introducon to Astrophysics, C. Bertulani, Texas A&M-Commerce 21